On the disconnection of a discrete cylinder by a biased random walk

Abstract

We consider a random walk on the discrete cylinder (ℤ/Nℤ)^d×ℤ, d≥3 with drift N^−dα in the ℤ-direction and investigate the large N-behavior of the disconnection time T_N^disc, defined as the first time when the trajectory of the random walk disconnects the cylinder into two infinite components. We prove that, as long as the drift exponent α is strictly greater than 1, the asymptotic behavior of T_N^disc remains N^2d+o(1), as in the unbiased case considered by Dembo and Sznitman, whereas for α<1, the asymptotic behavior of T_N^disc becomes exponential in N.